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SENTENCE MEANING AND PROPOSITIONAL CONTENT CHAPTER OUTLINE 1.1 Proposition 1.2 Propositional content 1.3 Notational representation of propositions 1.4 Truth functionality 1.5 Sentence types and their meaning The truth value of a proposition should be distinguished from the truth conditions of a sentence e.g. Mary married a rich man. Truth value of a proposition: The proposition can be either true or false (hence a twovalued proposition). Truth conditions of the sentence: Mary must be a woman. Mary is of a marriageable age.
CHAPTER SENTENCE MEANING AND PROPOSITIONAL CONTENT CHAPTER OUTLINE 1.1 Proposition 1.2 Propositional content 1.3 Notational representation of propositions 1.4 Truth functionality 1.5 Sentence types and their meaning 1.1 Proposition (Revisited) • A Proposition is defined as the invariant meaning expressed by a sentence, devoid of any modality e.g She is probably right • Proposition: She is right • Modality: varying from probable to impossible • In semantics, the letters ‘p, q, r’ are often used as symbols of propositions • Propositions involve in the meanings of not only declarative, but also interrogative and imperative sentences e.g Is she right? (You) be careful Truth-value vs Truth-conditions • The truth value of a proposition should be distinguished from the truth conditions of a sentence e.g Mary married a rich man • Truth value of a proposition: The proposition can be either true or false (hence a two-valued proposition) • Truth conditions of the sentence: Mary must be a woman Mary is of a marriageable age 1.2 Propositional content Propositional content Semantic roles Predicates & Arguments Propositional content • A proposition usually consists of (a) something which is named or talked about known as ARGUMENT (tham tố/tham thể) or entity, and (b) an assertion or predication made about the arguments expressed by the PREDICATE (vị từ) e.g The man bit the dog The dog bit the man • Predicate: BITE • Two arguments: MAN & DOG • The meaning of a sentence consists of the predicate, argument(s), and the role of each argument When we specify all these elements, we are talking about the propositional content of the sentence Arguments • Not all entities are arguments e.g It rained heavily • The arguments may fall into two sub-groups: participant and non-participant • Participants are those necessitated by the predication, and answer the question: Who does what to whom? • Non-participants are optional and answer the questions: why, when, where, how? e.g The woman hit the man (with a ruler) • There are three arguments: the woman, the man, ruler • In standard grammatical treatment, participant arguments surface as subject, direct or indirect object whereas non-participant arguments occur as adverbials Predicator - Predicate • A PREDICATE is any word, or sequence of words, which, in a given single sense, can function as the predicator of a sentence e.g hungry, in, asleep, hit, show, bottle: are predicates and, or, but, not: are not predicates • Predicate (vị từ / vị thể) and predicator (vị tố) are terms of quite different sorts The term ‘predicate’ identifies elements in the language system, independently of particular example sentences • The term ‘predicator’ identifies the semantic role played by a particular word (or groups of words) in a particular sentence • A simple sentence has one predicator, although it may well contain more than one instance of a predicate https://www.slideshare.net/AshwagAlhamid/unit-5-predicates-sv Degrees of predicates • The combination of predicate and arguments can be defined in terms of degree The DEGREE of a predicate is a number indicating the number of arguments it is normally understood to have in a simple sentence • A predicate of degree one (often called a one-place predicate) is used with one argument e.g asleep, beautiful • A prediate of degree two (often called a two-place predicate) is used with two arguments e.g kill, see • A predicate of degree three (often called a three-place predicate) is used with three arguments e.g give, make Arguments vs predicates • Arguments refer to entities while predicates deal with events, properties, attributes and states • Those individuals that are independent and can stand alone are arguments • Things like qualities, relations, actions and processes that are dependent and cannot stand alone are termed predicates e.g my computer break down, fast, new 10 Conjunction • The English words and and or correspond roughly to logical connectives Connectives provide a way of joining simple propositions to form complex propositions A logical analysis must state exactly how joining propositions by means of a connective affects the truth of the complex propositions so formed • Any number of individual well-formed formulae can be placed in a sequence with the symbol “&” between each adjacent pair in the sequence The result is a complex wellformed formula • E.g The three simple formulae: jGREETm (John greeted Mary), jHUGm (John hugged Mary) and jKISSm (John kissed Mary) can be joined together to form: (jGREETm) & (jHUGm) & (jKISSm) 30 Conjunction • Truth table for & p q p&q T T T T F F F T F F F F She left and I stayed 31 Conjunction • This operation generates a composite proposition, symbolized as p & q, which is true if and only if both p and q are true For example: • In a situation in which Henry died and Terry resigned is both true, then Henry died and Terry resigned is true • In a situation where Henry died is true, but Terry resigned is false, then Henry died and Terry resigned is false • Where Henry died is false, but Terry resigned is true, then Henry died and Terry resigned is also false • Where Henry died and Terry resigned are both false, then Henry died and Terry resigned is also false 32 Disjunction • Any number of well-formed formulae can be placed in a sequence with the symbol V between each adjacent pair in the sequence The result is a complex well-formed formula • For example, from the simple propositions: hHERE: Harry is here cDUTCHMAN Charlie is a Dutchman A single complex formula can be formed: (hHERE) V (cDUTCHMAN) • More examples: Dorothey saw Bill or Alan Either John or Peter has used my computer 33 Disjunction • Truth table for V p q pVq T T T T F T F T T F F F You can get there (either) by train or by bus 34 Disjunction • Disjunctions creates a composite proposition: p V q, which is true a If and only if either p or q is true and b If and only if both p and q are true • In a situation in which Henry died and Terry resigned is both true, then (Either) Henry died or Terry resigned is true • Where Henry died is true, but Terry resigned is false, then (Either) Henry died or Terry resigned is true • Where Henry died is false, but Terry resigned is true, then (Either) Henry died or Terry resigned is also true • Where Henry died and Terry resigned are both false, then (Either) Henry died or Terry resigned is also false 35 Implication • The logical connective symbolized by → corresponds roughly to the relation between an ‘if’ clause and its sequel in English The linking of two propositions by → forms what is called a conditional • If Alan is here, Clive is a liar 36 Material Implication • Truth table p q pq T T T T F F F T T F F T If Mary invited John, he will go 37 Material Implication • This operation creates a composite proposition whereby p → q (p implies q) p → q is true if and only if: (a) Both p and q are true (b) Both p and q are false (c) p is false and q is true • It is false if p is true and q is false For example If she has married him, they are honeymooning in HL now • This composite proposition can be true if (1) it is true that she has married him and they are honeymooning in HL now, or (2) it is false that she has married him and it is also false that they are honeymooning in HL now If it is true that she has married him but it is false that they are honeymooning in HL now, then the proposition is false because p does not imply q The last case is when she has not married him but they are honeymooning in HL now, which is found by most people to be paradoxical 38 • Entailment p is true and q is necessarily true (i.e true in all possible worlds) If dogs are mammals, they are animals • Implicature The truth of q can be inferred from p in certain contexts in which p is made If Trang’s cellphone is on, she must be writing a message or making a phone 39 Equivalence The conjunction of two implications • Truth table for ≡ p q p≡q T T T T F F F T F F F T If the horse is a mammal, the shark is a fish 40 Equivalence • The logical connective symbolized by ≡ expresses the meaning ‘if and only if’ in English The linking of two propositions by ≡ produces what is called a ‘biconditional’ • E.g The meaning of “John is married to Mary if and only if Mary is married to John” could be represented as: (jMARRYm) ≡ (mMARRYj) • The biconditional connective is aptly named because it is equivalent to the conjunction of two conditionals, one ‘going in each direction’ Inother words, there is a general rule: p ≡ q is equivalent to (p → q) & (q →p) 41 Negation • The connective ~ used in propositional logic is paraphrasable as English ‘not’ Strictly speaking, ~ does not CONNECT propositions, as (&) and (V) ~ is prefixed to the formula for a single proposition, producing its negation ~ is sometimes called the ‘negation operator’, rather than ‘negation connective’ • E.g Alice didn’t sleep can be represented as ~ aSLEEP Clare is not married to Bill = ~cMARRYb 42 Negation • Truth table for ~ P ~P T F F T I love you 43 THE END ... must be a woman Mary is of a marriageable age 1.2 Propositional content Propositional content Semantic roles Predicates & Arguments Propositional content • A proposition usually consists of (a)... attributes and states • Those individuals that are independent and can stand alone are arguments • Things like qualities, relations, actions and processes that are dependent and cannot stand alone... proposition: p V q, which is true a If and only if either p or q is true and b If and only if both p and q are true • In a situation in which Henry died and Terry resigned is both true, then (Either)